Hopf algebras for ternary algebras

Abstract

We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincaré-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order three. It turns out that these variables are the variables which generate the three-exterior algebra.

21 pages

References in corpus (7)

• Modeling Multiple M2's
• Poincaré and sl(2) algebras of order 3
• Kinematical superalgebras and Lie algebras of order 3
• n-ary associative algebras, cohomology, free algebras and coalgebras
• Ternary algebras and groups
• Non-trivial extension of the Poincaré algebra for antisymmetric gauge fields
• Cubic extentions of the Poincaré algebra

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• Hidden quartic symmetry in N=2 supersymmetry
• Hom-coassociative ternary coalgebras and infinitesimal bialgebras
• Braided quantum mechanics and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework
• (Co)associative $3$-ary (co)algebras and infinitesimal bialgebras: construction and main properties